On the weak invariance principle for strictly stationary sequences in some Besov spaces

In this paper, we investigate the weak invariance principle in some Besov spaces for strictly stationary sequences. In particular, we provide sufficient conditions on the moments and the dependence of a strictly stationary sequence to guaranty the weak invariance principle in these Besov spaces. Several dependence structures are considered, including projective conditions, $\alpha$-dependent, $\rho$-mixing, and $\tau$ -dependent conditions. The main tools in the proofs are martingale approximation and deviation inequalities for weakly dependent sequences. Our results cover a large range of strictly stationary sequences that were not treated previously in the literature. Moreover, the presented conditions are sharp.